Score : $500$ points

问题陈述

这是一个交互式问题（你的程序通过输入和输出与裁判进行交互）。

你给定一个正整数 $N$ 和整数 $L$ 和 $R$，使得 $0 \leq L \leq R < 2^N$。裁判有一个隐藏的序列 $A = (A_0, A_1, \dots, A_{2^N-1})$，由0到99之间的整数组成。

你的目标是找出 $A_L + A_{L+1} + \dots + A_R$ 除以100的余数。然而，你不能直接知道序列 $A$ 中元素的值。相反，你可以向裁判提出以下问题：

• 选择非负整数 $i$ 和 $j$，使得 $2^i(j+1) \leq 2^N$。设 $l = 2^i j$ 和 $r = 2^i (j+1) - 1$。询问 $A_l + A_{l+1} + \dots + A_r$ 除以100的余数。

设 $m$ 为确定任何序列 $A$ 的 $A_L + A_{L+1} + \dots + A_R$ 除以100的余数所需的最少问题数量。你需要在 $m$ 个问题内找到这个余数。

以上为大语言模型 kimi 翻译，仅供参考。

Problem Statement

This is an interactive problem (where your program interacts with the judge via input and output).

You are given a positive integer $N$ and integers $L$ and $R$ such that $0 \leq L \leq R < 2^N$. The judge has a hidden sequence $A = (A_0, A_1, \dots, A_{2^N-1})$ consisting of integers between $0$ and $99$, inclusive.

Your goal is to find the remainder when $A_L + A_{L+1} + \dots + A_R$ is divided by $100$. However, you cannot directly know the values of the elements in the sequence $A$. Instead, you can ask the judge the following question:

• Choose non-negative integers $i$ and $j$ such that $2^i(j+1) \leq 2^N$. Let $l = 2^i j$ and $r = 2^i (j+1) - 1$. Ask for the remainder when $A_l + A_{l+1} + \dots + A_r$ is divided by $100$.

Let $m$ be the minimum number of questions required to determine the remainder when $A_L + A_{L+1} + \dots + A_R$ is divided by $100$ for any sequence $A$. You need to find this remainder within $m$ questions.

Constraints

• $1 \leq N \leq 18$
• $0 \leq L \leq R \leq 2^N - 1$
• All input values are integers.

Input and Output

This is an interactive problem (where your program interacts with the judge via input and output).

First, read the integers $N$, $L$, and $R$ from Standard Input:

$N$ $L$ $R$

Then, repeat asking questions until you can determine the remainder when $A_L + A_{L+1} + \dots + A_R$ is divided by $100$. Each question should be printed in the following format:

$?$ $i$ $j$

Here, $i$ and $j$ must satisfy the following constraints:

• $i$ and $j$ are non-negative integers.

• $2^i(j+1) \leq 2^N$

The response to the question will be given in the following format from Standard Input:

$T$

Here, $T$ is the answer to the question, which is the remainder when $A_l + A_{l+1} + \dots + A_r$ is divided by $100$, where $l = 2^i j$ and $r = 2^i (j+1) - 1$.

If $i$ and $j$ do not satisfy the constraints, or if the number of questions exceeds $m$, then $T$ will be -1.

If the judge returns -1, your program is already considered incorrect. In this case, terminate the program immediately.

Once you have determined the remainder when $A_L + A_{L+1} + \dots + A_R$ is divided by $100$, print the remainder $S$ in the following format and terminate the program immediately:

$!$ $S$

Notes

• At the end of each output, print a newline and flush Standard Output. Otherwise, the verdict may be TLE.
• If your output is malformed or your program exits prematurely, the verdict will be indeterminate.
• After printing the answer, terminate the program immediately. Otherwise, the verdict will be indeterminate.

Example

Here is an example for $N=3$, $L=1$, $R=5$, and $A=(31, 41, 59, 26, 53, 58, 97, 93)$. In this case, $m=3$, so you can ask up to three questions.

Input	Output	Description
3 1 5		First, the integers $N$, $L$, and $R$ are given.
	? 0 1	Ask the question with $(i, j) = (0, 1)$.
41		$l=1$ and $r=1$, so the response is the remainder when $A_1=41$ is divided by $100$, which is $41$. The judge returns this value.
	? 1 1	Ask the question with $(i, j) = (1, 1)$.
85		$l=2$ and $r=3$, so the response is the remainder when $A_2 + A_3 = 85$ is divided by $100$, which is $85$. The judge returns this value.
	? 1 2	Ask the question with $(i, j) = (1, 2)$.
11		$l=4$ and $r=5$, so the response is the remainder when $A_4 + A_5 = 111$ is divided by $100$, which is $11$. The judge returns this value.
	! 37	The answer is $37$, so print this value.